Higher Mathematics

323 Pages·2002·6.693 MB·English

by  I. Suvorov

I. SUVOROV
higher
mathematics
TEXTBOOK FOR TECHNICAL SCHOOLS
PEACE PUBLISHERS MOSCOW
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I. SUVOROV
HIGHER
MATHEMATICS
TEXTBOOK FOR TECHNICAL SCHOOLS
Translated from the Hassian
f>V
M. V. OAK
Translation Editor
GEOlH.i; YANKOVSKY
PEACE PUBLISHERS MOSCOW
First Publish)
Second Printin
CONTENTS
A. DASIC ANALYTIC GEOMETRY IN THE PLANE
Chapter I. Method of Coordinates
Sec. 1. The Coordinates of a Point . . . . 11
Sec. 2. The Sum of Two Directed Segments 13
Sec. 3. The Distanco Between Two Points . . . 14
Sec. 4. Dividing a Segment in a Given Ratio 17
Sec. 5. The Angle of a Straight Line with the Axis . . 18
Sec. 6. Conditions for Parallelism and Perpendicularity 20
Chapter //. The Straight Line
Sec. 7. A Straight Line as a Locus......................................... 22
Sec. 8. Equation of a Straight Line (Slope-Intercept Form) . . . . 23
Sec. 0. General Form of the Equation of a Straight Line and Its
Special Cases....................................................................................... 25
Sec. 10. Equation of a Straight Line (Intercept Form)........................ 27
Sec. 11. Solved Examples 27
Sec. 12. Construction of a Straight Line When Its Equation Is Given 29
Sec. 13. The Point of Intersection of Two Straight Lines . . . 3 0
Sec. 14. Equation of a Straight Lino Passing Through the Point
(xi, y]) in a Given Direction 33
Sec. 15. Equation of a Straight Lino Passing Through Two Given
Points (x|t y,) and (x2, y2) . . . 34
Sec. 16. The Anglo Bolwccn Two Straight Lines 35
Chapter ///. Quadric Curves
Sec. 17. Equations of the Circle.................................................................. 39
Sec. 18. Solved Examples............................................................................... 40
Sec. 16. The Circle as a Quadric Curvo . . . . . 42
Sec. 20. Ellipso.................................................................................................... 44
Sec. 21. The Equation of an Ellipso.............................................................. 45
Sec. 22. Investigating the Form of an Ellipso from Its Equation . . . 46
Sec. 23. Plotting an Ellipso........................................................................... 48
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Sec. 24. Relationship Between the Ellipse and the Circle................... 49
Sec. 25. Eccentricity of an Ellipse.............................................................. 51
Sec. 20. Hyperbola................................................................................................. 51
Sec. 27. The Equation of the Hyperbola................................................... 52
Sec. 28. Investigating the Forms of the Hyperbola from Its Equation 53
Sec. 29. Plotting a Hyperbola....................................................................... 54
Sec. 30. Asymptotes of the Hyperbola...................................................... 56
Sec. 31. Eccentricity of a Hyperbola.......................................................... 58
Sec. 32. Equilateral Hyperbola....................................................................... 59
Sec. 33. Solved Examples on the Ellipse and Hyperbola..................... 59
Sec. 34. Parabola ................................................................................................ 60
Sec. 35. Equation of a Parabola.................................................................. 61
Sec. 36. Investigating the Forms of the Parabola from Its Equation 62
Sec. 37. Plotting a Parabola........................................................................... 03
Sec. 38. Formulas for the Transformation of Coordinates..................... 64
Sec. 39. Equation of the Parabola in Parallel Translation of the
Coordinate Axes................................................................................ 66
Sec. 40. Equation of an Equilateral Hyperbola Referred to the
Asymptotes . . . 67
Sec. 41. Solved Examples............................................................................... 68
Sec. 42. Quadric Curves as Conic Sections.................................................. 70
B. ELEMENTS OF DIFFERENTIAL CALCULUS
diopter IV. Theory of Limits
Sec. 43. Absolulo Value and Its Properties............................................. 73
Sec. 44. Infinitely Small Quantity (Infinitesimal)..................................... 75
Sec. 45. Variable Quantities, Hounded and rnhonnded......................... 76
Sec. 46. Basic Properties of Infinitesimals.................................................. 76
Sec. 47. Infinitely Large Quantity............................................................... 78
Sec. 48. Relationship Between Infinitely Small and Infinitely Largo
Quantities................................................................................................ 79
Sec. 49. The Limit of a Variable Quantity.............................................. 80
Sec. 50. t'lOumctrical Representation of a Number, Variable, and
Limit......................................................................................................... 83
Sec. 51. Relationship Between a Variable, Its Limit, and an Infini­
tesimal ..................................................................................................... 86
Sec. 52. A Variable Can Have (inly One Limit.......................................... 86
Sec. 53. The Limit of an Algebraic Sum .................................................. 87
Sec. 54. The Limit of a Product................................................................. 87
Sec. 55. The Limit of a Quotient................................................................ 88
Sec. :>li. Tho Limit of a Rational Algebraic Expression........................... 90
Sec. 57. The Sign of a Variable and Its Limit......................................... 91
Sec. 58. Conditions for the Existence of a Limit of a Variable . . . . 91
Sec. 59. On the Limit of a Quotient of Infinitesimals............................. 92
Sec. OH. Examples in Finding Limits . . . 92
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Chapter V. Function and Its Continuity
Sec. 61. Argument and Function................................................................... 95
Sec. 62. General Designation of a Function.............................................. 97
Sec. 63. Graphical and Analytical Representation of a Function . . . 93
Sec. 64. Graph of a Function..................................................................... 100
Sec. 65. Increment of the Argument and Function................................ 102
Sec. 66. The Limit of a Function at a Finite Point................................ 104
Sec. 67. The Limit of a Function When x —>- oo.................................... 106
Sec. 68. Some Observations......................................................................... 107
Sec. 69. Continuity of a Function............................................................. 108
Sec. 70. Another Expression for the Condition of Continuity of a
Function................................................................................................ 112
Sec. 71. Testing a Function for Continuity................................................ 113
Sec. 72. The Properties of Functions Continuous at a Point . . . . 113
Chapter VI. Derivative Function
Sec. 73. Linear Function, Its Rate of Change.......................................... 115
Sec. 74. Examples in Finding Rates of Change..................................... 116
Sec. 75. Derivative Function...................................................................... 119
Sec. 76. Tangent to a Curve........................................................................... 122
Sec. 77. Geometrical Meaning of a Derivative.......................................... 124
Sec. 78. Relationship Between Differentiability and Continuity of
' a Function.....................................................................'.................. 127
Chapter VII. Derivatives of Elementary Functions
Sec. 79. Preliminary Remarks...................................................................... 128
Sec. 80. The Derivative of a Constant...................................................... 128
Sec. 81. The Derivative of a Power.......................................................... 128
Sec. 82. The Derivative of tho Product of a Constant and a Function 130
Sec. 83. Tho Derivative of an Algebraic Sum of Functions..................... 131
Sec. 84. The Derivative of a Product of Functions................................. 132
Sec. 85. Tho Derivative of a Fraction.......................................................... 133
Sec. 86. Remarks......................................................... 136
Sec. 87. Tho Function of a Function.......................................................... 136
Sec. 88. The Derivative of a Function of a Function............................. 136
Sec. 89. Tho Limit of the Ratio of a Sino to an Arc............................. 139
Sec. 90. Derivatives of Trigonometric Functions..................................... HO
Sec. 91. Two Systems of Logarithms. Tho Number r. Changing from
One System of Logarithms to thoO ther....................................... 143
Sec. 92. The Derivative of a Logarithm...................................................... 145
Sec. 93. Monotonic Functions...................................................................... 148
Sec. 94. The Derivative of an Inverse Function..................................... 149
Sec. 95. The Derivative of on Exponential Function............................. |50
Sec. 96. The Derivative of Any Power...................................................... 151
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Sec. 97. Derivatives of Inverse Trigonometric Functions 151
Sec. 98. Derivatives of Second and Higher Orders . . . 153
Chapter VIII. Studying Functions with the Aid of Their Derivatives
Sec. 99. How to Determine Whether a Function Increases, Decreases
or Is Constant..................................................................................... 154
Sec. 100. Extreme Value Problems................................................................ 157
Sec. 101. Maximum and Minimum of a Function...................................... 159
Sec. 102. A Test for Extremes........................................................................ 160
Sec. 103. Procedure for Finding Extremes......................... . . . 162
Sec. 104. Examples in Finding Extremes................................................... 162
Sec. 105. Second Derivative Test for Extreme Values.............................. 163
Sec. 106. Extreme Value Problems.................................................................... 167
Sec. 107. Maximum and Minimum of a Function at Points Where
the Derivative Has No Value....................................................... 170
Sec. 108. The Direction of Concavity of a Curve . . . . . . 171
Sec. 109. Points of Inflection........................................................................ 172
Sec. 110. Constructing Graphs of Functions............................................... 173
Sec. 111. Mechanical Interpretation of the Second Derivative . . 174
Chapter IX. Differential
Sec. 112. Comparing InGnilosimals.............................................................. 176
Sec. 113. Differential of a Function............................................................. 177
Sec. 114. The Differential of an Argument. The Derivative as a Ratio
of Differentials................................................................................ 179
Sec. 115. Applying the Concept of Differential to Approximate
Calculations........................................................................................ 181
C. ELEMENTS OF INTEGRAL CALCULUS
Chapter X. Indefinite Integral
Sec. 116. Integration as the In verso of Differentiation......................... 185
Sec. 117. Tho Indefinite Integral as an Expression of the Aggregate
of Antiderivatives of a Given Function...................................... 187
Sec. 118. Properties of an Indefinite Integral . . . . . 189
Sec. 119. Integration by Formulas.............................................. . 190
Sec. 120. Integration by Substitution . . . . . . . . 191
Sec. 121. Standard Integrals and Their Uses............................................... 195
Sec. 122. Integration of Powers of sin *, cos x, tan x, cot r ............. 201
Sec. 123. ^ V\i2— x‘ <lx 203
Sec. 124. Remarks . . oqx
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